In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f : R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that for all x , y ∈ R d {\displaystyle x,y\in \mathbb {R} ^{d}} such that x {\displaystyle x} is majorized by y {\displaystyle y} , one has that f ( x ) ≤ f ( y ) {\displaystyle f(x)\leq f(y)} . Named after Issai Schur, Schur-convex functions are used in the study of majorization.
A function f is 'Schur-concave' if its negative, −f, is Schur-convex.
Properties
Every function that is convex and symmetric (under permutations of the arguments) is also Schur-convex.
Every Schur-convex function is symmetric, but not necessarily convex.1
If f {\displaystyle f} is (strictly) Schur-convex and g {\displaystyle g} is (strictly) monotonically increasing, then g ∘ f {\displaystyle g\circ f} is (strictly) Schur-convex.
If g {\displaystyle g} is a convex function defined on a real interval, then ∑ i = 1 n g ( x i ) {\displaystyle \sum _{i=1}^{n}g(x_{i})} is Schur-convex.
Schur–Ostrowski criterion
If f is symmetric and all first partial derivatives exist, then f is Schur-convex if and only if
( x i − x j ) ( ∂ f ∂ x i − ∂ f ∂ x j ) ≥ 0 {\displaystyle (x_{i}-x_{j})\left({\frac {\partial f}{\partial x_{i}}}-{\frac {\partial f}{\partial x_{j}}}\right)\geq 0} for all x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}}holds for all 1 ≤ i , j ≤ d {\displaystyle 1\leq i,j\leq d} .2
Examples
- f ( x ) = min ( x ) {\displaystyle f(x)=\min(x)} is Schur-concave while f ( x ) = max ( x ) {\displaystyle f(x)=\max(x)} is Schur-convex. This can be seen directly from the definition.
- The Shannon entropy function ∑ i = 1 d P i ⋅ log 2 1 P i {\displaystyle \sum _{i=1}^{d}{P_{i}\cdot \log _{2}{\frac {1}{P_{i}}}}} is Schur-concave.
- The Rényi entropy function is also Schur-concave.
- x ↦ ∑ i = 1 d x i k , k ≥ 1 {\displaystyle x\mapsto \sum _{i=1}^{d}{x_{i}^{k}},k\geq 1} is Schur-convex if k ≥ 1 {\displaystyle k\geq 1} , and Schur-concave if k ∈ ( 0 , 1 ) {\displaystyle k\in (0,1)} .
- The function f ( x ) = ∏ i = 1 d x i {\displaystyle f(x)=\prod _{i=1}^{d}x_{i}} is Schur-concave, when we assume all x i > 0 {\displaystyle x_{i}>0} . In the same way, all the elementary symmetric functions are Schur-concave, when x i > 0 {\displaystyle x_{i}>0} .
- A natural interpretation of majorization is that if x ≻ y {\displaystyle x\succ y} then x {\displaystyle x} is less spread out than y {\displaystyle y} . So it is natural to ask if statistical measures of variability are Schur-convex. The variance and standard deviation are Schur-convex functions, while the median absolute deviation is not.
- A probability example: If X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} are exchangeable random variables, then the function E ∏ j = 1 n X j a j {\displaystyle {\text{E}}\prod _{j=1}^{n}X_{j}^{a_{j}}} is Schur-convex as a function of a = ( a 1 , … , a n ) {\displaystyle a=(a_{1},\dots ,a_{n})} , assuming that the expectations exist.
- The Gini coefficient is strictly Schur convex.
See also
References
Roberts, A. Wayne; Varberg, Dale E. (1973). Convex functions. New York: Academic Press. p. 258. ISBN 9780080873725. 9780080873725 ↩
E. Peajcariaac, Josip; L. Tong, Y. (3 June 1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic Press. p. 333. ISBN 9780080925226. 9780080925226 ↩